Theorem of the trident

Set the Trident:
The three red lines are the same length: .
${\displaystyle |DA|=|DI|=|DC|}$

The theorem of the trident (after the Russian terms лемма о трезубце (literally: theorem about the trident ) and теорема трилистника (literally: theorem of the trillium )) is a statement from elementary geometry that describes a property of the circumference and the inner circle of a triangle .

In a triangle, let the center of its inscribed circle and the intersection of (bisector in ) with its circumference, then the following equivalent statements apply: ${\displaystyle ABC}$ ${\displaystyle I}$ ${\displaystyle D}$ ${\displaystyle BI}$ ${\displaystyle B}$

• The distances , and are the same length, that is . ${\displaystyle DA}$ ${\displaystyle DI}$ ${\displaystyle DC}$ ${\displaystyle |DA|=|DI|=|DC|}$
• ${\displaystyle A,I,C}$lie on a circle whose center is. In particular, the center of the circle lies through and on the circumference of the triangle . ${\displaystyle D}$ ${\displaystyle A,I}$ ${\displaystyle C}$ ${\displaystyle ABC}$

Web links

• Eric W. Weisstein : Incenter-Excenter Circle . In: MathWorld (English).
• A Property of Circle Through the Incenter on cut-the-knot.org
• Midpoints of the Lines Joining In- and Excenters on cut-the-knot.org

Individual evidence

1. ↑ Р. Н. Карасёв, В. Л. Дольников, И. И. Богданов, А. В. Акопян: Задачи для школьного математического кружка (PDF), p. 4.
2. ↑ 6. Лемма о трезубце (PDF) СУНЦ МГУ им. М. В. Ломоносова - школа им. А.Н. Колмогорова. October 29, 2014.
3. ↑ И. А. Кушнир: Это открытие - золотой ключ Леонарда Эйлера. (PDF) Ф7 (Теорема трилистника), p. 34; Proof on p. 36